Tight Frame Graphs Arising as Line Graphs

Dual multiplicity graphs are those simple, undirected graphs that have a weighted Hermitian adjacency matrix with only two distinct eigenvalues.  From the point of view of frame theory, their characterization can be restated as which graphs have a representation by a tight frame.  In this paper, we classify certain line graphs that are tight frame graphs and improve a previous result on the embedding of frame graphs in tight frame graphs

Classification of Generalized Multiresolution Analyses

We discuss how generalized multiresolution analyses (GMRAs), both classical and those defined on abstract Hilbert spaces, can be classified by their multiplicity functions $m$ and matrix-valued filter functions $H$. Given a natural number valued function $m$ and a system of functions encoded in a matrix $H$ satisfying certain conditions, a construction procedure is described that produces an abstract GMRA with multiplicity function $m$ and filter system $H$. An equivalence relation on GMRAs is defined and described in terms of their associated pairs $(m,H)$. This classification system is applied to classical examples in $L^2 (\mathbb R^d)$ as well as to previously studied abstract examples.Comment: 18 pages including bibliograp

The Spark of Symmetric Matrices Described by a Graph

We investigate the sparsity of null vectors of real symmetric matrices whose off-diagonal pattern of zero and nonzero entries is described by the adjacencies of a graph. We use the definition of the spark of a matrix, the smallest number of nonzero coordinates of any null vector, to define the spark of a graph as the smallest possible spark of a corresponding matrix. We study connections of graph spark to well-known concepts including minimum rank, forts, orthogonal representations, Parter and Fiedler vertices, and vertex connectivity.Comment: 20 pages, 2 figure

Nordhaus–Gaddum problems for power domination

A power dominating set of a graph G is a set S of vertices that can observe the entire graph under the rules that (1) the closed neighborhood of every vertex in S is observed, and (2) if a vertex and all but one of its neighbors are observed, then the remaining neighbor is observed; the second rule is applied iteratively. The power domination number of G, denoted by gamma p(G), is the minimum number of vertices in a power dominating set. A Nordhaus-Gaddum problem for power domination is to determine a tight lower or upper bound on gamma p(G) + gamma p(G) or gamma p(G).gamma p(G), where G denotes the complement of G. The upper and lower Nordhaus-Gaddum bounds over all graphs for the power domination number follow from known bounds on the domination number and examples. In this note we improve the upper sum bound for the power domination number substantially for graphs having the property that both the graph and its complement are connected. For these graphs, our bound is tight and is also significantly better than the corresponding bound for the domination number. We also improve the product upper bound for the power domination number for graphs with certain properties

Zero forcing and power domination for graph products

The power domination number arose from the monitoring of electrical networks, and methods for its determination have the associated application. The zero forcing number arose in the study of maximum nullity among symmetric matrices described by a graph (and also in control of quantum systems and in graph search algorithms). There has been considerable effort devoted to the determination of the power domination number, the zero forcing number, and maximum nullity for specific families of graphs. In this paper we exploit the natural relationship between power domination and zero forcing to obtain results for the power domination number of tensor products and the zero forcing number of lexicographic products of graphs. In addition, we establish a general lower bound for the power domination number of a graph based on the maximum nullity of the matrices described by the graph. We also establish results for the zero forcing number and maximum nullity of tensor products and Cartesian products of certain graphs

Racial differences in systemic sclerosis disease presentation: a European Scleroderma Trials and Research group study

Objectives. Racial factors play a significant role in SSc. We evaluated differences in SSc presentations between white patients (WP), Asian patients (AP) and black patients (BP) and analysed the effects of geographical locations.Methods. SSc characteristics of patients from the EUSTAR cohort were cross-sectionally compared across racial groups using survival and multiple logistic regression analyses.Results. The study included 9162 WP, 341 AP and 181 BP. AP developed the first non-RP feature faster than WP but slower than BP. AP were less frequently anti-centromere (ACA; odds ratio (OR) = 0.4, P < 0.001) and more frequently anti-topoisomerase-I autoantibodies (ATA) positive (OR = 1.2, P = 0.068), while BP were less likely to be ACA and ATA positive than were WP [OR(ACA) = 0.3, P < 0.001; OR(ATA) = 0.5, P = 0.020]. AP had less often (OR = 0.7, P = 0.06) and BP more often (OR = 2.7, P < 0.001) diffuse skin involvement than had WP.AP and BP were more likely to have pulmonary hypertension [OR(AP) = 2.6, P < 0.001; OR(BP) = 2.7, P = 0.03 vs WP] and a reduced forced vital capacity [OR(AP) = 2.5, P < 0.001; OR(BP) = 2.4, P < 0.004] than were WP. AP more often had an impaired diffusing capacity of the lung than had BP and WP [OR(AP vs BP) = 1.9, P = 0.038; OR(AP vs WP) = 2.4, P < 0.001]. After RP onset, AP and BP had a higher hazard to die than had WP [hazard ratio (HR) (AP) = 1.6, P = 0.011; HR(BP) = 2.1, P < 0.001].Conclusion. Compared with WP, and mostly independent of geographical location, AP have a faster and earlier disease onset with high prevalences of ATA, pulmonary hypertension and forced vital capacity impairment and higher mortality. BP had the fastest disease onset, a high prevalence of diffuse skin involvement and nominally the highest mortality

Generalized filters, the low-pass condition, and connections to multiresolution analyses

AbstractWe study generalized filters that are associated to multiplicity functions and homomorphisms of the dual of an abelian group. These notions are based on the structure of generalized multiresolution analyses. We investigate when the Ruelle operator corresponding to such a filter is a pure isometry, and then use that characterization to study the problem of when a collection of closed subspaces, which satisfies all the conditions of a GMRA except the trivial intersection condition, must in fact have a trivial intersection. In this context, we obtain a generalization of a theorem of Bownik and Rzeszotnik